Hopf bifurcation and the Hopf fibration

Abstract

We present techniques for studying the local dynamics generated by an equivariant Hopf bifurcation. We show that under general hypotheses we can expect the formation of a branch of attracting invariant spheres which capture all the local dynamics. In addition, using the Hopf fibration, we show that the limit cycles generated by the Hopf bifurcation are determined by zeros of a vector field defined on complex projective space. We show how to compute these zeros and illustrate our methods with examples of Hopf bifurcations for the dihedral groups of order six and eight and the orthogonal groups.